Atmos. Chem. Phys., 4, 51–63, 2004
www.atmos-chem-phys.org/acp/4/51/
SRef-ID: 1680-7324/acp/2004-4-51
Atmospheric
Chemistry
and Physics
Source-receptor matrix calculation with a Lagrangian particle
dispersion model in backward mode
P. Seibert and A. Frank
Institute of Meteorology and Physics, University of Natural Resources (BOKU), Wien, Austria
Received: 17 June 2003 – Published in Atmos. Chem. Phys. Discuss.: 29 August 2003
Revised: 12 December 2003 – Accepted: 10 January 2004 – Published: 23 January 2004
Abstract. The possibility to calculate linear-source receptor relationships for the transport of atmospheric trace substances with a Lagrangian particle dispersion model (LPDM)
running in backward mode is shown and presented with
many tests and examples. This mode requires only minor
modifications of the forward LPDM. The derivation includes
the action of sources and of any first-order processes (transformation with prescribed rates, dry and wet deposition, radioactive decay, etc.). The backward mode is computationally advantageous if the number of receptors is less than
the number of sources considered. The combination of an
LPDM with the backward (adjoint) methodology is especially attractive for the application to point measurements,
which can be handled without artificial numerical diffusion.
Practical hints are provided for source-receptor calculations
with different settings, both in forward and backward mode.
The equivalence of forward and backward calculations is
shown in simple tests for release and sampling of particles,
pure wet deposition, pure convective redistribution and realistic transport over a short distance. Furthermore, an application example explaining measurements of Cs-137 in Stockholm as transport from areas contaminated heavily in the
Chernobyl disaster is included.
and instead of a grid volume average it could be the value
at a measurement station. The source could be a point, area
or volume source acting during a specified time interval in a
specific location.
The s–r relationship can be linear or nonlinear. In the nonlinear case, it is defined by the partial derivative ∂y/∂x at a
given state of the system. This paper deals with linear s–r
relationships, because standard Lagrangian particle dispersion models cannot simulate nonlinear chemical reactions.
However, all the other processes occurring during the atmospheric transport of trace substances are linear: advection,
diffusion, convective mixing, dry and wet deposition, and radioactive decay. First-order chemical reactions, where the
reaction rates can be prescribed, are also linear. In the linear
case, the s–r relationship reduces to the simple expression
y/x. Thus, linear s–r relationships can be calculated easily
with any dispersion model M : x→y.
In typical applications, there are many receptor elements
and/or many sources. We will denote them by vectors y and
x, respectively. The s–r relationships can be written as a s–r
matrix (also abbreviated as SRM) M whose elements mil are
defined by
mil =
yl
.
xi
1 Introduction
The source-receptor (s–r henceforth) relationship is an important concept in air quality modelling. It describes the sensitivity of a “receptor” element y to a “source” x. The receptor could be, for example, the average concentration of a
certain atmospheric trace substance in a given grid cell during a given time interval. It could also be a deposition value,
Correspondence to: P. Seibert
(www.boku.ac.at/imp/envmet/seibert mail.html)
© European Geosciences Union 2004
In the case of gridded data, the temporal and all the three
spatial dimensions are combined into each of the indices
(i=1, .., I for the sources, and l=1, .., L for the receptors).
This notation is in agreement with inverse modelling studies. In the remaining sections of this paper, receptors are
specified as mixing ratios or concentrations and as such denoted with symbols χ and c. Sources are specified as mass
flux densities or masses and denoted by q̇ or Q, respectively.
Thus, the s–r relationship can be denoted by ∂χ /∂ q̇ or similar expressions.
52
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
The advantage of knowing M is that for a given source
vector, the resulting receptor values can be obtained by a
simple matrix-vector multiplication, avoiding the evaluation
of the whole numerical model of the transport and dispersion
processes. This is useful for scenario considerations. Furthermore, the SRM contains detailed budget information; it
describes which receptors are affected by a specific source
element (by the corresponding column of M) and which
sources are contributing to a specific receptor (by the row
of M) corresponding to the receptor). The SRM gives a
quite comprehensive description of a transport and dispersion problem. Its determination is therefore also an important
intermediate step for solving optimisation problems, such
as the minimisation of environmental impact under defined
abatement costs, or inverse modelling to determine sources
from given measurements.
The determination of the SRM is much more costly in
terms of computing than a single simulation with a given
source configuration because it has to be either rerun for
each source element, or (more efficiently but also requiring more computer memory) a separate species has to be
tracked for each source element (in forward mode). In the
case of I >L (more source elements than receptor elements),
receptor-oriented methods are more efficient than forward
methods, and vice versa. These receptor-oriented methods
are known as adjoint or backward methods. They need a
number of simulations or species equal to the number of
receptors (whereas forward methods need as many simulations or species to be tracked as there are sources). For
the case of Eulerian models, many authors have pointed this
out and applied adjoint Eulerian models for such tasks (e.g.
Uliasz (1983), Uliasz and Pielke (1992), Robertson and Persson (1992), Giering (1999), Hourdin and Issartel (2000)).
For Lagrangian particle models, Thomson (1987) and Flesch
et al. (1995) have shown that the Lagrangian particle model
is basically self-adjoint (except the sign of advection). Flesch
et al. (1995) introduced backward integration of such a model
as a tool to determine so-called footprints for measurements
of micro-meteorological fluxes. Footprint modelling including the backward methodology has become an established
tool in the micrometeorological community for interpretation
of both fluxes and scalar quantities and the planning of measurements (Schmid, 2002; Kljun et al., 2002).
Seibert (2001) introduced this concept in air pollution
modelling and for arbitrary volume sources. However, all
these papers were limited to conservative tracers, without
considering sinks or any other first-order processes.
The present papers shows that a Lagrangian particle dispersion model can be used in a backward-running, receptororiented mode to determine s–r relationships including any
such first-order processes. The concept is valid for source
and receptor volumes of arbitrary shape and location. We
apply it to problems ranging from mesoscale to global. The
Atmos. Chem. Phys., 4, 51–63, 2004
derivation is followed by sample calculations illustrating
practically the equivalence between forward and backwardmode calculations and an application example. Especially
Test 3 and the associated Fig. 1 may serve to help understand
the principle of s–r relationships and their calculation by forward or backward calculations.
Methods presented here have been integrated into the
model Flexpart1 (Stohl et al., 1998), which was also used
for the sample calculations. This model is presently used by
about 15 groups all over the world and it is freely available.
We feel therefore that it is not inappropriate to include some
remarks on a more technical level which are related specifically to this model.
To avoid misunderstandings, we would like to point out
that the Lagrangian particle dispersion model should not be
confused with other types of Lagrangian models such as standard trajectory models or box models tied to such standard
trajectories. The LPDM is a fully three-dimensional model
including the mean horizontal and vertical wind and threedimensional turbulence.
2
2.1
Theory
Forward simulations with an LPDM
In the forward mode, particles are released in prescribed
source regions, and transported with the mean and a stochastic turbulent velocity field. They carry a mass depending on
source strength and particle release rate which can be altered
by processes such as dry and wet deposition or decay. Gridded concentrations are calculated by summing up the masses
of all the particles in a grid cell, and dividing the sum by the
cell’s volume. More refined formulations, used for example in the model Flexpart (Stohl et al., 1998) may apply kernels (Haan, 1999) to determine gridded and especially singlereceptor point concentrations.
In the backward mode, used for receptor-oriented modelling, the same formalism and computer model are applied,
but the particle trajectories are integrated backward in time,
using a negative time step. However, these particles are only
a means to determine the trajectories and to probe the processes experienced by substances transported. Furthermore,
the approach is based on mass mixing ratios rather than mass
concentrations, as this is a conservative quantity.
1 more information at http://www.forst.tu-muenchen.de/EXT/
LST/METEO/stohl/flexpart.html
www.atmos-chem-phys.org/acp/4/51/
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
53
7
2.2 Lagrangian, receptor-oriented view of transport and
able 2. Results ofdispersion
release and sampling test.
type of calculation
s–r relationship [s]
In the following, a mathematical description of the transport
analytical
43,200
and dispersion process in the atmosphere including sources
forward and linear processes (such as, e.g.43,233
deposition or decay) is
backward given in a Lagrangian framework 43,233
with a receptor-oriented
Number of particles:
view. The1000
result is compared to the formalism employed in
Synchronisation
time
interval: particle
300 s dispersion models.
typical
Lagrangian
We consider the individual change of the mass mixing ratio χ of an atmospheric trace substance. As we are in a Laable 3. Results of wet scavenging test.
grangian frame of reference, there are no advective changes
and χ is affected only by sources
q̇ or by any
type of calculation
s–r relationship
[s] linear processes
which
are
proportional
to
χ.
Nonlinear
processes are
analytical
2668
not
considered
in
this
paper.
The
sources
q̇
are
given
as mass
forward
2666
per
volume
and
time.
Thus,
backward
2664
Number of particles: 1000
dχ (t)
1
Synchronisation time
s (t),
= interval:
q̇(t) 300
+ α(t)χ
(1)
dt
ρ(t)ª¬«l­„®°¯²±šF³W´‚µ=«l±g¶ · mm š› h
Scavenging parameters:
Precipitation rate [mm h š¸› ]: 1.9 + 0.1 INT (time / 300 s)
where α is the net decay or build-up rate “constant” and t is
time.
This ordinary differential equation can be solved, yielding
.2 Test 2: Wet
scavenging
∗
the mixing ratio at time t and location r :
t
his test is theχ(r
same
as Test 1, except
that a strong wet
∗
, t) = χ0 exp −
α[r(t 0 ), t 0 ] dt 0 +
cavenging is switched on with a prescribed,
linearly vary0
Z t
Z t
0 ), t 0 ]
q̇[r(t
ng precipitation
rate.
In the available
versions
of
00
00
00 Flexpart,
exp
−
α[r(t
),
t
]
dt
dt 0 ,
(2)
0 ), tall
0 ] particles0 regardless of their height.
wet scavenging
acts on
0 ρ[r(t
t
hus, the true s–r relationship could be calculated analyti0 ≤ t} is the back trajectory arriving at point
{r(t 0 ), tof
ally. For awhere
description
the wet scavenging algorithm in
∗
at timeand
t, and
χ0 is the
initial mixing
ratio (background)
lexpart, seer Stohl
Seibert
(2002).
Furthermore,
tests
at t=0.
were conducted for stationary precipitation fields with conLet us write
as large-scale
an abbreviation
ective precipitation
only,
precipitation only, or
mixed precipitation, allwith
a
total
cloud
cover
of 1. In all
Z t
Fig. 1. Example of the forward, source-oriented mode (top) and the
0
00
00
00
hese tests, differences
most),at few
p(t ) = exp were
− atα[r(t
] dt seconds (Table 3). (3) Fig. backward,
receptor-oriented
mode
(bottom) calculations
1. Example
of the forward,
source-oriented
mode performed
(top) and the
Z
t0
for Testreceptor-oriented
3: realistic short-range
of calculations
a conservativeperformed
tracer.
backward,
modetransport
(bottom)
Calculations were performed for 11 October 2000 over the Baltic
3: realistic short-range transport of a conservative tracer.
call it thescenario
transmission
function becausetransport
it determines the for Test
.3 Test 3:and
Realistic
for short-distance
Sea and its surroundings (coast lines and borders (as of 1986) are
were performed for 11 October 2000 over the Baltic
fraction of material which is transmitted along a trajectory to Calculations
shown by thin lines). The light square indicates the release areas
and
its mode.
surroundings
lines and
borders
1986]
the receptor. Let us further write in abbreviated form q̇(t 0 ) Sea in
each
Sampling(coast
is performed
on the
whole [as
grid,ofbut
only are
n this test, and
particles
werethereleased
theit denotes
same grid
volume
ρ(t 0 ) with
meaningin
that
q̇ and
ρ along the shown
thin oflines).
The
lightcan
square
indicates
the release
areas
the by
sample
the bold
square
be used
for comparison
of the
s in the previous
tests,
however,
theyEquation
were sampled
antrajectory
as explained
above.
(2) then in
reads
in each
mode. Sampling
is performed
on the of
whole
grid,
but only
forward-mode
and backward-mode
calculation
the s–r
relation◦ . The parameter shown is the
ship. Theoflength
of each
squarecan
is 1be
ther grid volume two degrees Zto the south-east, during the
the sample
the bold
square
used for comparison of the
t q̇(t 0 )
s–r
relationship
[s],
calculated
with
the
mode represented
by green
forward-mode
and
backward-mode
calculation
of the s–r
relationame 24 h asχthe
release.
The
co-ordinates
of
the
grid
centres
0
0
(r ∗ , t) = χ0 p(0) +
p(t
)
dt
.
(4)
squares
in
Fig.
2.
0)
¹
ship.
The
length
of
each
square
is
1
.
The
parameter
shown
is the
ρ(t
were 57 ¨ N, 20 ¨ E and 59 ¨ N, 18 ¨ E.
0 Real meteorological fields
s–r
relationship
[s],
calculated
with
the
mode
represented
by
green
ECMWF analysis, 1 ¨ horizontal resolution, 60 layers, every
and (4) are valid for instantaneous mixing squares in Figure 2.
h) were usedEquations
as input,(2)and
three different types of tracers
Thus, we need to average χ in time. If the averaging time
ratios, affected by turbulent fluctuations. The mean mixing
were simulated:
conservative,
short-lived (‘radioactive noble
ratios should be obtained as ensemble averages. In practiexceeds well the time scale of the turbulent fluctuations of
as’, half-life
d), and ameasured
tracer subject
to wet at
scavenging
χ calculations
(and it does so
in trace
monitoring), the
temcal0.5
applications,
concentrations
monitoring sta- ward
modes
forsubstance
the s–r relationship
in general.
aerosol’, scavenging
properties
as ininTest
test was
tions represent
nearly a point
space2).
butThis
an average
in time.
poral average of the instantaneous values is the same as the
In the forward mode, particles are released from the area of
arried out for the days 1–20 October 2000. Convection was
witched offwww.atmos-chem-phys.org/acp/4/51/
(note that this refers only to vertical redistriution through convection, convective precipitation is taken
om ECMWF analysis and is always considered). In this
ase, fluctuations of the concentrations occur as a result of
the light square, which is the source in the s–r relationship
Atmos.
Chem.
Phys.,are
4, 51–63,
2004 by
considered in this test, and
tracked
as they
transported
the atmospheric flow. The difference of this mode to regular dispersion modelling is only that results have been scaled
54
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
temporal average of the respective ensemble means (χ̄=hχ i,
where the overbar indicates temporal averaging and h i the
ensemble averaging). Similar considerations hold for aircraft
measurements which are typically line integrals or satellite
measurements representing volume integrals. For the mean
(time-integrated) mixing ratio χ̄ we obtain
χ̄ (t1 , t2 ) =
1
t2 − t1
Z
t2
1
t2 − t1
Z
t2
χ(r ∗ , t) dt =
t1
t
Z
χ0 (t) p(t, 0) +
t1
0
q̇(t, t 0 )p(t, t 0 ) 0
dt
dt, (5)
ρ(t, t 0 )
where a second time variable t appears in q̇, ρ, and p to
denote trajectories arriving at different times t.
We now introduce a discretisation. Arrival time t is discretised into J time slots each represented by one back trajectory. The trajectories arrive at equal intervals between t1
and t2 and are designated by index j . Space is gridded with
index i. Furthermore, we need a discretisation of t 0 , denoted
by the index n. This leads to
1 X X X q̇in
0
χ̄ ≈ χ0 p(0) +
pj n 1tij n
J j i n ρin
(6)
with 1tij0 n being the residence time of trajectory j in the
spatio-temporal grid cell (i, n). Note that the integration
along the trajectory (a line integral in the joint time-space dimension) has been replaced by a double sum, one over time
and one over space, with the residence time being not only
the discrete representation of dt but also the indicator of the
trajectory movement. As the source function q̇/ρ depends
only on the time when the back trajectory passes but not on
the time when it arrives at the receptor, Eq. (6) can be rearranged to (writing = instead of ≈ for simplicity)
"
#
X X q̇in 1 X
χ̄ = χ0 p(0) +
(pj n 1tij0 n ) .
ρ
J
in
n
i
j
The transmission-corrected residence times, yielding the s–r
relationships according to Eq. (8), can be calculated easily
with a Lagrangian particle dispersion model running backward in time. As discussed in in Sect. 2.1, regular LPDMs
generate particles at source locations, follow their forward
trajectories, and let different processes such as deposition act
on the mass attached to the computational particle. By reversing the sign of the advection, the trajectories can easily
be calculated backward. The LPDM Flexpart (Stohl et al.,
1998) has this capability already built in. It required a few
other modifications in the code, allowing input and output
to occur in reverse temporal direction, etc. The first-order
processes that have been summed up in the function α remain the same as in a forward calculation, in the sense that
the mass will be, e.g. reduced by decay whether going backward or forward. Due to the linear nature of the term αχ , the
direction of the time-stepping does not matter. If the backward calculation reverses the sign of the time step instead of
the advection velocities (as does Flexpart), the integration of
processes related to α must be carried out with the absolute
value of the time step.
Turbulence is represented in LPDMs by a stochastic process with a prescribed pdf for the turbulent wind components (or their random parts, if a model with autocorrelation
is used). Let P (vi ) be the probability density function for
the velocity component vi . In order to calculate back trajectories, this probability must be replaced by P (−vi ) (Flesch
et al., 1995). If only a symmetric pdf is implemented, for example a Gaussian distribution as in Flexpart, no code modification is required. An integration with a negative time step
will give the correct backward trajectories. However, the
cloud convection scheme in Flexpart has different probabilities for upward and downward movements and thus needed
to be adapted (see Sect. 2.6).
The statistics of the turbulence do not depend on the direction of a trajectory. In mathematical terms, this means
that the Lagrangian equation of transport in a turbulent fluid,
in the absence of asymmetric random motions as discussed
above, is self-adjoint with the exception of a change of sign
in the mean-flow advection (Flesch et al., 1995).
(8)
Let us now see how the standard output of a model like
Flexpart can be used to extract these transmission-corrected
residence times. Flexpart writes out mass concentrations c̄,
averaged over a grid element (index i) and a time interval
1T (index n). They are computed as
In the case of α=0 and thus p=1, the s–r relationship for
a source expressed as mass mixing ratio (q̇/ρ) is the average residence time of the back trajectories in the grid cell of
the source under consideration, in agreement with the earlier
derivation for this specific case (Seibert, 2001). In the general case, the residence times are to be “corrected” with the
transmission function p.
Atmos. Chem. Phys., 4, 51–63, 2004
Computation of transmission-corrected residence times
with a Lagrangian particle dispersion model
(7)
We are now able to calculate the s–r relationship:
0
∂ χ̄
1 X pj n 1tij n
=
.
∂ q̇in
J j
ρin
2.3
c̄in
1
=
Vi N ∗
n+N
X∗
J
X
µj n∗ fij n∗ ,
(9)
n∗ =n+1 j =1
www.atmos-chem-phys.org/acp/4/51/
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
where N ∗ is the number of samples taken within 1T , and
the time-dependent particle mass µj n∗ is
(10)
µj n∗ = µ0 pj n∗
with µ0 being the initial mass associated with a particle
(which is a constant for each release in Flexpart). fij n∗ is
a function that determines if and how much a particle contributes to a given grid cell. In the simplest case, fij n∗ would
be 1 if trajectory j is in grid cell i at time step n∗ and 0 otherwise; in Flexpart, it is used with a kernel formulation which
distributes the mass over four grid elements as a measure to
counteract the numerical errors due to the limited number of
trajectories. Combining Eqs. (9) and (10), we obtain for the
LPDM output:
c̄in =
µ0
Vi N ∗
n+N
X∗
J
X
Table 1. Implementation scheme for different concentration units
in an LPDM. “mass” refers to mass units (kg m -3 s-1 for the source,
kg m -3 for the receptor), “mix” to mixing ratio units (s-1 for the
source, dimensionless for the receptor). Given are in the column
“release” the factor with which the initial mass of a particle has to
be multiplied, and in the column “sampling” the factor to multiply
particles’ masses with at the moment of sampling. Note that in
backward simulations the release takes place at the receptor and the
sampling at the source. The respective units of the s–r relationship
are given in the last column. The first value (C:) is for sources given
as rates per time (and volume in the case of mass), the second one
(I:) for instantaneous local sources (for example, a mass in units of
kg). They are the same for forward and backward calculations!
(11)
pj n∗ fij n∗ .
n∗ =n+1 j =1
The calculation of the s–r relationship with the LPDM
requires, of course, that the temporal discretisation of the
source as introduced in the previous section matches the time
intervals used for averaging the output in the LPDM. For a
numerical representation, we can adopt the same strategy as
in Flexpart and divide the time interval of interest into N ∗
small time slices, using the same function f for each of these
intervals. Thus, Eq. (8) becomes
1T
∂ χ̄
= ∗
∂ q̇in
N
55
n+N
X∗
J
pj n∗ fij n∗
1X
.
J j =1 ρin∗
n∗ =n+1
(12)
Considering that J is independent of n∗ , we may write
∂ χ̄
Vi 1T
=
∂ q̇in
J µ0
cin
ρin
=
Vi 1T
µtot
cin
ρin
(13)
with µtot as abbreviation for the total initial mass released in
the time interval between t1 and t2 . (cin /ρin ) differs from the
standard output of the LPDM as given in Eq. (11), c̄in , only
in the density by which the contributions of the single particles in an output grid cell have to be divided. This division
is necessary because we have expressed the source in mass
units. If, instead, the source would be specified as alteration
rate of the mixing ratio, the standard output c̄in could be used
directly. Note that in this case the dimension of the r.h.s. is
time. In any case, it is obvious that the standard LPDM provides a way to obtain s–r relationships by running it backward in time with very few modifications.
A key assumption made to arrive at this result is that particles carry mixing ratios rather than masses. This is equivalent to the change from ∇ · ρχ v to v · ∇ρχ between forward
and adjoint Eulerian transport models (Elbern and Schmidt,
www.atmos-chem-phys.org/acp/4/51/
1999). At both ‘ends’ of a simulation, at the source and at
the receptor, mass units may be desired. This is true for forward as well as for backward simulations. The conversion
between both forms is accomplished by multiplication or division with the local air density. As density can vary within
a source or receptor volume, this conversion is best done
particle-wise. Work on this paper and associated tests as well
as ongoing practical applications have shown that keeping
maximum flexibility in an LPDM code with respect to mixing ratio versus mass units is desirable. Table 1 indicates how
to set up a model for all the possible combinations. However,
additional scalings are needed if the s–r relationship is to be
derived from forward calculations and defined in the same
way as done here (see Eq. 18).
2.4
Practical considerations
For gases, volume mixing ratios are a more popular unit than
mass mixing ratios. The conversion is simple: multiplication with the ratio of the molecular mass of air to that of the
gaseous trace substance or vice versa.
Another widespread practice is to report mass concentrations at standard temperature and pressure (STP). In this
case, Table 1 can be applied with local air density ρ replaced
by the air density at STP. As this is a constant, it is possible,
for example, to do a backward simulation for receptor units
of mixing ratio, and to divide the resulting s–r relationship by
ρ STP later in order to use it in conjunction with measurements
given in STP concentrations.
Atmos. Chem. Phys., 4, 51–63, 2004
56
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
Another issue is that often we would be interested in an
area source at ground level. To accomplish this, let us consider any area A and a vertical layer adjacent to the ground of
thickness d as the source volume. Diluted over this layer, an
area source q̇A (in kg m-2 s-1) would correspond to a volume
source (in kg m-3 s-1) q̇=q̇A d −1 . Thus,
∂ χ̄
1 ∂ χ̄
=
.
∂ q̇A
d ∂ q̇
(14)
Obviously, as ∂ χ̄/∂ q̇A contains the (density-weighted) residence time in the layer, it will, for well-mixed conditions and
a layer not too deep, be proportional to d, so that ∂ χ̄ /∂ q̇A is
independent of d. The condition of being well-mixed is usually fulfilled within the boundary layer after some travel distance, except for a strongly depositing substance under stable
conditions. To obtain good numerical stability, one should
not choose the depth of the lowest layer too small, and except for the above-mentioned circumstances it can be chosen to be the typical minimum mixing height expected in the
model. It is also interesting to note that the residence time in
this layer of a single trajectory being reflected on the ground
with vertical velocity w is 2d/w. Thus, the s–r relationship
for a conservative tracer (p=1) and an area source is
J
2d fj
∂ρχ
1 X
J ∗ −1
=
=2
w ,
∂ q̇A
J d j =1 wj
J
(15)
twice the fraction of trajectories impinging on the ground
weighted with the inverse of their vertical velocity, a conclusion already drawn by Flesch et al. (1995).
To complete the practical formulae, the s–r relationship for
a source in kg s-1 (point source, one-grid cell source), having
units of s kg-1, would be
∂ χ̄
1T
=
Vi ∂ q̇in
µtot
cin
,
ρin
(16)
and for an instantaneous point source (or the total source for
a given time interval) in kg, having units of kg-1,
∂ χ̄
1
=
1T Vi ∂Qin
µtot
cin
.
ρin
(17)
Instead of expressing ambient concentrations and sources
in kilogramme, one can always substitute that by Becquerel
if radioactivity is under consideration.
Flexpart and other LPDMs allow not only to calculate
gridded concentrations but also concentrations at point receptors, by means of suitable kernels sampling particles in a
certain neighbourhood of the receptor. This can be used to
obtain s–r relationships for point sources if the (candidate)
source locations are known.
Atmos. Chem. Phys., 4, 51–63, 2004
To convert gridded concentrations resulting from standard
forward simulations into s–r relationships with the dimension
of time, a transformation is necessary. For source and receptor both in mass units, it reads as follows (for other cases,
factors as indicated in Table 1 need to applied):
∂ χ̄
1Ts Vs
=
c̄,
∂ q̇in
µtot
(18)
where 1Ts is the time during which the (constant!) source is
acting, Vs is its volume and c̄ is the concentration in a grid
cell as delivered by the forward run. For Flexpart, the above
value must be multiplied with the factor 10−12 , because results are scaled by that factor.
2.5
Dry and wet deposition
Source-receptor relationships are also of interest with respect
to receptors representing accumulated deposition. The deposition fluxes are obtained by linear operators on the the
concentration field (in an LPDM, on the masses of particles). Therefore it is easy to extend the methods presented
to this case. In an efficient implementation, pseudo-sources
with strengths varying as a function of time and space, proportional to the deposition velocity or wash-out coefficients,
would be used. This has not yet been implemented in Flexpart.
2.6
Convection
Most Lagrangian particle dispersion models do not simulate
the effects of moist (i.e., cloud) convection. However, a parameterisation for convective transports has been introduced
recently into Flexpart (Seibert et al., 2002). The scheme redistributes particles from their initial level to a randomly chosen destination level. The transition probabilities are given
in discrete form in a so-called redistribution matrix which is
calculated from the temperature and humidity profiles. Because convection manifests as concentrated updrafts with a
high vertical velocity and weak compensating subsidence occupying a larger area, the redistribution matrix is not symmetric. In a backward run, particles must be redistributed
from the destination level back to the initial level. Thus,
the transposed redistribution matrix has to be used. This
means that the probability of a particle of having arrived at
its present level from another level is considered.
Another issue which requires attention is the part of
the scheme which calculates the redistribution matrix. A
one-dimensional convection model (Emanuel and ŽivkovićRothman, 1999), designed for use in a prognostic meteorological model, is run for each grid column containing particles. The model is integrated forward in time for a few hours,
calculating Eulerian tracer tendencies due to the action of
www.atmos-chem-phys.org/acp/4/51/
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
convection, from which the redistribution matrix is derived.
Tendencies of the grid-scale temperature and humidity (taken
from the meteorological input to Flexpart) enter this simulation, and these tendencies must be forward in time, as the
convection submodel must always run forward in time, even
in an LPDM backward simulation. The convection subroutines are called only once for each set of meteorological input fields, thus typically every 3 or 6 h, in the middle of the
time between the two fields (this has been changed in Flexpart recently). If a sampling or release period is not distributed symmetrically around the time when convection is
calculated, differences between forward and backward mode
modes will arise. They are an unavoidable consequence of
the coarse temporal discretisation for the convection, which,
however, is necessary because the convection simulation is
so time-consuming.
3
Test calculations
Source-receptor relationships have been calculated both in
forward and backward mode for a number of test cases. The
purpose of these test calculations was to illustrate the concept and to have empirical support for the correctness of the
derivations presented in the first part of this paper. During
this work it turned out that these tests – comparison between
forward and backward simulations as well as calculations for
which the result can be analytically derived – is also a very
useful validation check for a model. In Flexpart, we found
a few bugs and model limitations in this process, mainly related to the numerical implementation of wet scavenging and
to the release and sampling of particles. Most of these bugs
would not affect standard applications in a significant way.
The tests are described here in the order of increasing complexity. They were carried out in final form with Flexpart
Version 5; unless otherwise mentioned, bugs and limitations
found were corrected for the tests. The next version of Flexpart will include all the improvements worked out during
these tests. More details will be provided through the Flexpart home page on the world-wide web (cf. Footnote 1).
Presently, particles are released in standard forward mode
of Flexpart equally distributed throughout the vertical extension of the source. It would be more realistic to release them
with the same profile as air density, but again, as long as the
layer is not too deep, the difference is not a big one.
3.1
Test 1: Release and sampling
In this test, particles representing a conservative tracer are
released during one day in a certain volume which is at
the same time the sampling volume. It was chosen to be
1◦ ×1◦ ×500 m large. All velocities (mean wind and turbuwww.atmos-chem-phys.org/acp/4/51/
57
Table 2. Results of release and sampling test.
type of calculation
analytical
forward
backward
Number of particles: 1000
Synchronisation time interval: 300 s
s–r relationship [s]
43,200
43,233
43,233
lence) are set to zero, and the sampling kernel (attributing
the mass of each particle to the four surrounding grid cells)
was switched off so that particles are attributed completely
to the cell where they are located. Thus, the only processes
present in the model were release and sampling of particles,
and any errors are due to the discretisation of these processes.
(Therefore we can work with a relative small number of particles.) The sampling took place during the same 24 h in which
the source was active. Thus, the analytical s–r relationship
is the mean residence time of the particles in the volume,
43 200 s. Test results and some additional characteristics are
reported in Table 2. There is no difference between forward
and backward mode. The small difference (less than 1 per
mille) between numerical and analytical result is caused by
an inaccurate temporal discretisation of release and sampling
of particles, affecting the first and last intervals only.
3.2
Test 2: Wet scavenging
This test is the same as Test 1, except that a strong wet
scavenging is switched on with a prescribed, linearly varying precipitation rate. In the available versions of Flexpart,
wet scavenging acts on all particles regardless of their height.
Thus, the true s–r relationship could be calculated analytically. Scavenging is implemented in Flexpart as a scavenging
rate which is a nonlinear function of the precipitation intensity (Stohl and Seibert, 2002). Furthermore, tests were conducted for stationary precipitation fields with convective precipitation only, large-scale precipitation only, or mixed precipitation, all with a total cloud cover of 1. In all these tests,
differences were at most a few seconds (Table 3).
3.3
Test 3: Realistic scenario for short-distance transport
In this test, particles were released in the same grid volume
as in the previous tests, however, they were sampled in another grid volume two degrees to the south-east, during the
same 24 h as the release. The co-ordinates of the grid centres were 57◦ N, 20◦ E and 59◦ N, 18◦ E. Real meteorological
fields (ECMWF analysis, 1◦ horizontal resolution, 60 layers,
every 3 h) were used as input, and three different types of
tracers were simulated: conservative, short-lived (“radioactive noble gas”, half-life 0.5 d), and a tracer subject to wet
Atmos. Chem. Phys., 4, 51–63, 2004
58
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
200
s−r relationsip difference FWD − BWD [s]
Table 3. Results of wet scavenging test.
5%
10%
type of calculation
s–r relationship [s]
analytical
2668
forward
2666
backward
2664
Number of particles: 1000
Synchronisation time interval: 300 s
Scavenging parameters: a = 2 × 10−4 , b = 0.8 mm−1 h
Precipitation rate [mm h−1 ]: 1.9 + 0.1 INT (time / 300 s)
100
1%
0
−100
300 s, all on (normal Flexpart)
300 s, Flexpart without turbulence and kernel
300 s, 20 boxes, 400,000 particles
300 s, 500 boxes, 1,000,000 particles
900 s, 500 boxes, 1,000,000 particles
−200
0
1000
2000
3000
4000
s−r relationship FWD [s]
Fig. 2. Scatter plot of s–r relationships [s] for the realistic shortrange transport of a conservative tracer (Test 3). The differences between forward and backward calculation results are plotted against
the forward results. Each dot represents one day. Dashed lines indicate relative deviations between the two calculation modes. The
calculations were carried out with different time steps and particle numbers (partly released in sub-boxes of the source for more
homogeneous initial distribution) as indicated in the insert. All calculations except the “normal Flexpart” are without turbulence and
kernel and, unless otherwise indicated, with 40 000 particles.
scavenging (“aerosol”, scavenging properties as in Test 2).
This test was carried out for the days 1–20 October 2000.
Convection was switched off (note that this refers only to
vertical redistribution through moist convection, convective
precipitation is taken from ECMWF analysis and is always
considered). In this case, fluctuations of the concentrations
occur as a result of the stochastic nature of the LPDM.
Figure 1 illustrates the test as well as the forward and backward calculations modes for the s–r relationship in general.
In the forward mode, particles are released from the area of
the light square, which is the source in the s–r relationship
considered in this test, and tracked as they are transported by
the atmospheric flow. The difference of this mode to regular dispersion modelling is only that results have been scaled
according to Eq. (18). In the backward mode, particles are
released from the area of the receptor in the s–r relationship
(which is marked by the bold square in the forward-mode
picture and the light square in the backward-mode picture)
and tracked backward in time, to their regions of origin. In
each calculation mode, only one sampling volume, indicated
by the bold square, may be compared with the other mode.
Atmos. Chem. Phys., 4, 51–63, 2004
Figure 2 compares the s–r relationships calculated in forward and backward mode with different set-ups of the model.
We notice that stochastic effects of turbulence produce deviations between both calculation modes. In one set-up, all
turbulent components were set to zero. However, even then
there are deviations. It was found that they can be diminished by using a large number of particles, and ensuring a
homogeneous initial distribution by dividing the source into
separate boxes. However, even then some deviations remain,
and they decrease when shorter time steps are used. They are
an expression of the inherent uncertainties of the trajectory
calculation caused by the need to interpolate wind fields to
particle positions.
This was corroborated by further tests. With artificial, constant wind fields no deviations occurred. One-dimensional
trajectories were calculated with the numerical scheme of
Flexpart in an analytical, nonlinear wind field. In the first
case, velocities were calculated directly from the analytical
formula at each position and in the second case they were
interpolated from values at a regular grid. For time steps going towards zero, errors were found to go to zero as expected
only in the first case, but not in the second one.
This is a further example that the forward-backward comparison is a useful model test in general, and it might even be
used as to estimate the total interpolation error in the results.
No differences in behaviour, as far as equality between forward and backward simulations is concerned, were found between species.
3.4
Test 4: Idealised convective redistribution
An idealised test was devised for the cloud convection
scheme, assuming complete mixing through moist convection. The redistribution matrix for the convection was defined
analytically so that it would lead to complete vertical mixing
within one step. The vertical levels used in this matrix were
ECMWF levels from a real case, and 50 000 particles were
used. Tests were conducted with instantaneous, local sources
and results are scaled to be valid for a one square-metre column. This test serves not only as a test for the backward
www.atmos-chem-phys.org/acp/4/51/
Seibertcalculation
and A. Frank:
Source-receptor
matrix
calculation
with amodel
Lagrangian particle dispersion9model
e-receptor P.matrix
with
a Lagrangian
particle
dispersion
fwd_mass-mass
been applied in a study related
surement network established
ystem for the Comprehensive
bert and Frank, 2001), and for
easurements of long-range polosphere (Stohl et al., 2002).
Source height [km a.g.l.]
n Figure 3. Apart from minor
by the stochastic redistribution
ion scheme, analytical and nue. It is left to the reader to reof the curves and how they are
ward calculation modes and the
eceptor.
20
15
10
5
0
0
.5
1.0
s-r relationship
[10^-4]
fwd_mass-mix
analytical 0.5 km
numerical 0.5 km
analytical 9.5 km
numerical 9.5 km
20
15
10
5
0
20
15
10
5
0
.5
1.0
s-r relationship
[10^-4]
fwd_mix-mass
15
10
5
0
.5
1.0
s-r relationship
[10^-4]
bwd_mass-mix
1.5
analytical
numerical
analytical
numerical
20
15
10
5
0
1.5
analytical 0.5 km
numerical 0.5 km
analytical 9.5 km
numerical 9.5 km
20
0
1.5
Source height [km a.g.l.]
essure and à is the gravity ac-
bwd_mass-mass
Source height [km a.g.l.]
top 2ÁÃ
Receptor height [km a.g.l.]
_
¡ j Receptor height [km a.g.l.]
m
H
Receptor height [km a.g.l.]
ion for convenience. For a col-
59
20
15
0
.5
1.0
s-r relationship
[10^-4]
bwd_mix-mass
1.5
analytical
numerical
analytical
22-Apr-03 20:51:56
numerical
22-Apr-03 20:35:
10
5
Source height [km a.g.l.]
Receptor height [km a.g.l.]
Receptor height [km a.g.l.]
e shall be presented here. Sta0
0
0
.5
1.0
1.5
0
.5
1.0
1.5
oactivity in Europe occasions-r relationship
[10^-4]
s-r relationship
[10^-4]
fwd_mix-mix
bwd_mix-mix
137Cs since the Chernobyl disanalytical 0.5 km
analytical
20
20
is radioactive material comes
fwd_mass-mass
numerical 0.5 km
numerical
contaminated areas. We have
analytical
9.5
km
analytical
2015
15
22-Apr-03 20:51:56
22-Apr-03 20:35:
t this hypothesis for the 137Cs
numerical 9.5 km
numerical
15
10
10
2000 in Stockholm, Sweden.
10
–r relationships have been cal5
5
as receptors and the area con5
0
0
0 kBq m-2 according to Euro0
0
.5
1.0
1.5
0
.5
1.0
1.5
0
.5
1.0
1.5
). Wet and dry deposition were
s-r relationship [10^-4]
s-r relationship [10^-4]
s-r relationship [10^-4]
which are based on ECMWF
analytical
analytical
analytical
0.5 km0.5 km
numerical
0.5
km
on and better temporal resolunumerical 0.5 km
numerical
analytical
9.5 km9.5 km
analytical
analytical
ith 1 ¨ horizontal and 3 h temnumerical 9.5 km
22-Apr-03 20:35:
numerical 9.5 km
numerical22-Apr-03 20:51:56
ons were carried out for transe the receptors have a resoluFig. 3. Analytical (red lines) and numerical (black lines) s–r relaon the filter
scheme),
Fig.change
3. Analytical
(red lines) and
numerical
lines)convection
s–r relationships
in a 1 m2 air column. Units are
tionships
for(black
idealised
tests for
in aidealised
1 m2 airconvection
column. tests
Units
Table 1, 7
instantaneous
local source. The left column of graphs shows forward simulations with s–r relationships as a function of
y integratedaccording
over theto whole
d
are according to Table 1, instantaneous local source. The left colthe receptor
for two different source heights (thin solid: 0.5 km, heavy broken: 9.5 km). The right column shows backward simulations
a 1 ¨F© 1 ¨ grid
mesh height
was used,
umn
showsfor
forward
simulations
s–r(colours
relationships
as
with s–r relationships as a function ofof
thegraphs
source height
two different
receptorwith
heights
correspondingly).
Forward and backward
and receptor
grid boxes
was for these
a function
of heights.
the receptor height for two different source heights
calculations
should agree
two specific
ed out with 500 000 particles.
(thin solid: 0.5 km, heavy broken: 9.5 km). The right column
22-Apr-03 20:51:56
shows backward simulations with s–r relationships as a function of
orward and backward simulathe source height for two different receptor heights (colours corre22-Apr-03 20:51:56
22-Apr-03 20:35:1
s and downs of the observaspondingly). Forward and backward calculations should agree for
s of the contamination within
these two specific heights.
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Atmos. Chem. Phys., 4, 51–63, 2004
iations in resuspension coeffiror (approximated as the mining between 2 and 7 Bq m -3)
60
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
15
relationship for one source and the whole profile of receptors,
whereas each backward calculation is for one receptor and
the whole source profile. Two calculations were performed
for each mode, with a low-level and a high-level release. If
we denote the source height by z and the receptor height by
ζ , and the release in mass units by µ and in mixing ratio units
by ν, the analytical solutions for the s–r relationships read:
10
mass – mass:
∂c(z, ζ )
ρ(ζ )
=
∂µ
MA
(19)
mass – mix:
∂χ(z, ζ )
1
=
∂µ
MA
(20)
mix – mass:
∂c(z, ζ )
ρ(z)ρ(ζ )
=
∂ν
MA
(21)
mix – mix:
∂χ(z, ζ )
ρ(z)
=
∂ν
MA
(22)
Observed and modelled Cs−137 concentration
Contaminated areas − Stockholm, October 2000
fwd model: 0.51 + 1.00 m (r = 0.74)
bwd model: 0.05 + 0.99 m (r = 0.94)
obs error bars
observation
Cs−137 [uBq/m3]
20
5
0
8
9
10
11
12
13
14 15 16
day number
17
18
19 20
21
22
Observed and modelled Cs−137 concentration
Chernobyl − Stockholm, October 2000
fwd model: 1.37 + 37 m (r = 0.69)
bwd model: 0.50 + 45 m (r = 0.81)
obs error bars
observation
Cs−137 [uBq/m3]
20
where c and χ are the mass concentration and the mass mixing ratio as previously, and MA is the total mass of the air
within the column that is being completely mixed. The short
source and receptor unit characterisations from Table 1 have
been included with each equation for convenience. For a column of 1 m2,
15
10
MA = (p bottom − p top ) g −1 ,
5
0
where p is the atmospheric pressure and g is the gravity acceleration.
8
9
10
11
12
13
14 15 16
day number
17
18
19 20
21
22
Fig. 4. Time series of the observed (with error margins) and modelled concentration of Cs-137 at Stockholm for a part of October
2000. The model is a simple regression between observations and
s–r relationship, calculated in either forward or backward mode,
between Stockholm and the Chernobyl area. The regression formulae and resulting correlation coefficients are given in the legend;
both are determined without the measurement of October 13th (outlier). The top panel is for a source covering the contaminated areas
in Ukraine, Belorus and Russia as indicated in Fig. 5, the bottom
panel for a source area limited to the grid element containing the
Chernobly power plant (0.5◦ size).
method, but also as a test for the multipliers given in Table 1,
because of the strong differences in ambient pressure. We
don’t compare forward and backward calculations directly,
but rather we compare both of them with the analytical solution. This way of presentation illustrates also the different
nature of forward (source-oriented) and backward (receptororiented) calculations. Each forward calculation gives the s–r
Atmos. Chem. Phys., 4, 51–63, 2004
The test results are shown in Fig. 3. Apart from minor fluctuations which are caused by the stochastic redistribution of
the particles by the convection scheme, analytical and numerical s–r relationships agree. It is left to the reader to reflect
about the different shapes of the curves and how they are related to the forward or backward calculation modes and the
different units for source and receptor.
4
Application examples
The new method has already been applied in a study related
to the global radionuclide measurement network established
as a part of the monitoring system for the Comprehensive
Nuclear Test Ban Treaty (Seibert and Frank, 2001), and for
the interpretation of aircraft measurements of long-range pollutant transport in the free atmosphere (Stohl et al., 2002).
In addition, a small example shall be presented here. Stations monitoring ambient radioactivity in Europe occasionally measure elevated levels of 137Cs since the Chernobyl disaster. It appears likely that this radioactive material comes
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P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
61
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagra
from resuspension in heavily contaminated areas. We have
done some calculations to test this hypothesis for the 137Cs
monitoring data of October 2000 in Stockholm, Sweden.
Backward as well as forward s–r relationships have been calculated for these measurements as receptors. As the source,
we considered either the areas in Ukraine, Belorus and Russia contaminated with more than 100 kBq m-2 according to
European Communities et al. (2001), or the single 0.5◦ grid
cell containing the Chernobyl power plant. Wet and dry deposition were considered in the simulations which are based
on ECMWF analyses and – for precipitation and better temporal resolution – short-range forecasts, with 1◦ horizontal
and 3 h temporal resolution. The simulations were carried
out for transport times up to 7 days. While the receptors have
a resolution of 1 to 3 days (depending on the filter change
scheme), the source has been temporally integrated over the
whole 7 d transport time. As receptor, a 1◦ ×1◦ grid mesh
was used, and the height of both source and receptor grid
boxes was 300 m. Simulations were carried out with 500 000
particles.
ogy. M
the ECM
dionucl
by the S
Refere
Figure 4 (top) shows that both forward and backward simulations reproduce the major ups and downs of the observations. Strong inhomogeneities of the contamination within
the source areas, potential variations in resuspension coefficients and the measurement error (approximated as the minimum detectable activity, varying between 2 and 7 µBq m-3)
limit the achievable agreement. On some days, there is a considerable difference between forward and backward simulations which we would attribute to general modelling errors,
mainly due to interpolation. The sample of October 12/13
is an outlier. It might be caused by caesium picked up in
an area northwest of Stockholm (Fig. 5, bottom), at a distance of less than 100 km, contaminated with values in the
40-100 kBq m-2 range (European Communities et al., 2001).
Leaving this case out, the model based on the backward simulation results in a correlation coefficient of 0.94 and the
expected direct proportionality between s–r relationship and
concentration (regression constant 0.05). Interestingly, the
model based on the forward simulation performs worse (correlation coefficient of 0.74).
If only the area immediately around the reactor is considered as potential source (Fig. 4, bottom), the correlation coefficient drops for both the forward and the backward simulation, indicating that it is indeed more likely that the observed
caesium originates from resuspension than from ventings of
the sarcophagus in which the reactor has been enclosed. Replacing the larger source area with the small one so that both
source and receptor are small, the correlation coefficient for
the backward model drops much stronger than for the forward model, and the difference between forward an backward models is reduced (especially if considered in terms of
explained variance). This could be a hint that a small release
area for the computational particles combined with a large receptor area (backward simulation for contaminated areas) is
www.atmos-chem-phys.org/acp/4/51/
Fig. 5. Some examples of s–r relationships for the receptor at StockFig.
Some
examples
s–rbackward
relationships
forfor
thedifferent
receptor measureat Stockholm5. as
determined
inofthe
mode
holm
as
determined
in
the
backward
mode
for
different
measurement periods. The s–r relationship is for an area source and has
ment
relationshipareas
is for
an area
source and
has
units periods.
of s m-1-1. The
The s–r
contaminated
around
Chernobyl
considunits
m evaluation
. The contaminated
areas
around by
Chernobyl
considered of
forsthe
in Fig. 4 are
indicated
their perimeters
ered
evaluation
in Figure
areby
indicated
their
perimeters
and for
the the
Chernobyl
power
plant 4site
a black by
dot.
The
receptor
and
the
Chernobyl
power
plant
site
by
a
black
dot.
The
at Stockholm is marked by a square. Political borders are receptor
those of
at1986.
Stockholm
is markedgrid
by awith
square.
Political
those
of
A geographical
a mesh
size ofborders
10◦ hasare
been
over¹ has
1986.
A southwestern
geographical corner
grid with
mesh
been
over◦ N,size
laid; the
is ata 40
10◦ of
E. 10
The
figures
refer
to
laid;
the southwestern
corner
is attop:
40 ¹ ending
N, 10 ¹ E.
figures middle:
refer to
the sampling
periods as
follows:
onThe
9 October,
the
samplingbottom:
periods 13
as October
follows: (the
top:outlier).
ending on 9 Oct, middle: 11
11 October,
Oct, bottom: 13 Oct (the outlier).
additional motivation. His contributions to discussions helped to
clarify a number of issues in the development of the methodol-
Atmos. Chem. Phys., 4, 51–63, 2004
www.atmos-chem-phys.org/0000/0001/
Emanue
evalu
Atmo
Europea
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4264
Stohl,
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62
P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model
less susceptible to numerical errors than vice versa (forward
simulation for contaminated areas).
5
Outlook
The method introduced in this paper has the potential to supersede back trajectories for the quantitative interpretation of
atmospheric trace substance measurements. For more qualitative interpretation, LPDM backward calculations with a
trajectory-like compressed output as suggested by Stohl et al.
(2002) can be recommended. Our vision is that a background
monitoring station would be associated with an archive of
the s–r data for each measurement. To keep the archive at
a reasonable size, the source vector would probably be gridded only in space but not in time, with one or a few layers
vertically and a grid resolution which is finer near the station and coarser at longer distances. Together with a small
software tool and an emission data base, the modelled contributions of selectable source regions to the station could be
quickly calculated and compared to measurements. Deviations could be analysed in various ways, including inverse
modelling (see, e.g. Seibert, 1999). A map of the (2-d or
3-d) s–r relationship (directly or multiplied with emissions)
could be termed a “footprint” of that station in analogy to
the micro-meteorological flux “footprints”. We admit, however, that this wording is not ideal as, rather, emissions leave
a kind of “footprint” in the atmosphere, and not vice versa.
We suggest an alternative term: the s–r relationship as the
“field of view” of a station.
Of course, there are also other methods to calculate such
s–r climatologies. The advantage of the backward method is
obvious for a receptor-oriented view. Compared to an adjoint
Eulerian model, the advantage of the LPDM is that there is
no initial diffusion due to the release of the adjoint tracer into
a finite-size grid cell, which gives the LPDM an advantage as
a measurement station acts like a point source in a backward
simulation. Furthermore, long-range transport can be simulated more accurately as no artificial numerical diffusion is
present. Especially when filaments created by deformation
processes become small against the mesh size, Eulerian models reach their limits. There are two drawbacks of the LPDM:
it cannot simulate nonlinear chemistry, and for very high dilution ratios, the necessary trade-off between particle number
and computational requirements will lead to statistically uncertain results (therefore a coarser grid may be required).
6
Conclusions
A method has been derived and introduced with various examples allowing to calculate relationships of substances not
undergoing nonlinear chemical changes with a backwardAtmos. Chem. Phys., 4, 51–63, 2004
running Lagrangian particle dispersion model. This includes
situations with wet and dry deposition or any other firstorder sources and sinks. Receptor-oriented problems, especially related to point or line (aircraft) measurements, can be
treated efficiently and relatively accurately by this method.
The method has been tested extensively with the Flexpart
(Stohl et al., 1998) model, and useful extensions have been
implemented in this model based on experiences gained in
these tests. It was also found that many of these tests are
generally useful for testing model quality. Differences between forward and backward simulations in realistic conditions may be a useful tool to obtain one estimate of model
errors, which is quite important for a subsequent inversion to
derive sources.
Acknowledgements. This work was supported by the Austrian Science Fund (FWF) through project P12595 “Inverse modelling of
atmospheric trace constituents with Lagrangian models on the European scale”, and it is a contribution to the EUROTRAC-2 subprojects GLOREAM and GENEMIS-2. A part of the manuscript was
written during a stay of the first author at the Canadian Meteorological Centre (CMC), Montréal, Canada; the support of the Meteorological Service of Canada and the warm reception by the colleagues
at CMC is gratefully acknowledged. The continuous co-operation
with A. Stohl (Technical University of Munich, Germany), main author of Flexpart, was important for the integration of the backward
calculation mode into this model and provided additional motivation. His contributions to discussions helped to clarify a number
of issues in the development of the methodology. Meteorological
fields for test calculations were provided by the ECMWF in the
frame of the Special Project “SPATMOT”. Radionuclide monitoring data from Stockholm were kindly supplied by the Swedish Defense Research Agency (I. Vintersved).
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